#include "blaswrap.h"
#include "f2c.h"

/* Subroutine */ int dpbsvx_(char *fact, char *uplo, integer *n, integer *kd, 
	integer *nrhs, doublereal *ab, integer *ldab, doublereal *afb, 
	integer *ldafb, char *equed, doublereal *s, doublereal *b, integer *
	ldb, doublereal *x, integer *ldx, doublereal *rcond, doublereal *ferr,
	 doublereal *berr, doublereal *work, integer *iwork, integer *info)
{
/*  -- LAPACK driver routine (version 3.0) --   
       Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,   
       Courant Institute, Argonne National Lab, and Rice University   
       June 30, 1999   


    Purpose   
    =======   

    DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to   
    compute the solution to a real system of linear equations   
       A * X = B,   
    where A is an N-by-N symmetric positive definite band matrix and X   
    and B are N-by-NRHS matrices.   

    Error bounds on the solution and a condition estimate are also   
    provided.   

    Description   
    ===========   

    The following steps are performed:   

    1. If FACT = 'E', real scaling factors are computed to equilibrate   
       the system:   
          diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B   
       Whether or not the system will be equilibrated depends on the   
       scaling of the matrix A, but if equilibration is used, A is   
       overwritten by diag(S)*A*diag(S) and B by diag(S)*B.   

    2. If FACT = 'N' or 'E', the Cholesky decomposition is used to   
       factor the matrix A (after equilibration if FACT = 'E') as   
          A = U**T * U,  if UPLO = 'U', or   
          A = L * L**T,  if UPLO = 'L',   
       where U is an upper triangular band matrix, and L is a lower   
       triangular band matrix.   

    3. If the leading i-by-i principal minor is not positive definite,   
       then the routine returns with INFO = i. Otherwise, the factored   
       form of A is used to estimate the condition number of the matrix   
       A.  If the reciprocal of the condition number is less than machine   
       precision, INFO = N+1 is returned as a warning, but the routine   
       still goes on to solve for X and compute error bounds as   
       described below.   

    4. The system of equations is solved for X using the factored form   
       of A.   

    5. Iterative refinement is applied to improve the computed solution   
       matrix and calculate error bounds and backward error estimates   
       for it.   

    6. If equilibration was used, the matrix X is premultiplied by   
       diag(S) so that it solves the original system before   
       equilibration.   

    Arguments   
    =========   

    FACT    (input) CHARACTER*1   
            Specifies whether or not the factored form of the matrix A is   
            supplied on entry, and if not, whether the matrix A should be   
            equilibrated before it is factored.   
            = 'F':  On entry, AFB contains the factored form of A.   
                    If EQUED = 'Y', the matrix A has been equilibrated   
                    with scaling factors given by S.  AB and AFB will not   
                    be modified.   
            = 'N':  The matrix A will be copied to AFB and factored.   
            = 'E':  The matrix A will be equilibrated if necessary, then   
                    copied to AFB and factored.   

    UPLO    (input) CHARACTER*1   
            = 'U':  Upper triangle of A is stored;   
            = 'L':  Lower triangle of A is stored.   

    N       (input) INTEGER   
            The number of linear equations, i.e., the order of the   
            matrix A.  N >= 0.   

    KD      (input) INTEGER   
            The number of superdiagonals of the matrix A if UPLO = 'U',   
            or the number of subdiagonals if UPLO = 'L'.  KD >= 0.   

    NRHS    (input) INTEGER   
            The number of right-hand sides, i.e., the number of columns   
            of the matrices B and X.  NRHS >= 0.   

    AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)   
            On entry, the upper or lower triangle of the symmetric band   
            matrix A, stored in the first KD+1 rows of the array, except   
            if FACT = 'F' and EQUED = 'Y', then A must contain the   
            equilibrated matrix diag(S)*A*diag(S).  The j-th column of A   
            is stored in the j-th column of the array AB as follows:   
            if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;   
            if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).   
            See below for further details.   

            On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by   
            diag(S)*A*diag(S).   

    LDAB    (input) INTEGER   
            The leading dimension of the array A.  LDAB >= KD+1.   

    AFB     (input or output) DOUBLE PRECISION array, dimension (LDAFB,N)   
            If FACT = 'F', then AFB is an input argument and on entry   
            contains the triangular factor U or L from the Cholesky   
            factorization A = U**T*U or A = L*L**T of the band matrix   
            A, in the same storage format as A (see AB).  If EQUED = 'Y',   
            then AFB is the factored form of the equilibrated matrix A.   

            If FACT = 'N', then AFB is an output argument and on exit   
            returns the triangular factor U or L from the Cholesky   
            factorization A = U**T*U or A = L*L**T.   

            If FACT = 'E', then AFB is an output argument and on exit   
            returns the triangular factor U or L from the Cholesky   
            factorization A = U**T*U or A = L*L**T of the equilibrated   
            matrix A (see the description of A for the form of the   
            equilibrated matrix).   

    LDAFB   (input) INTEGER   
            The leading dimension of the array AFB.  LDAFB >= KD+1.   

    EQUED   (input or output) CHARACTER*1   
            Specifies the form of equilibration that was done.   
            = 'N':  No equilibration (always true if FACT = 'N').   
            = 'Y':  Equilibration was done, i.e., A has been replaced by   
                    diag(S) * A * diag(S).   
            EQUED is an input argument if FACT = 'F'; otherwise, it is an   
            output argument.   

    S       (input or output) DOUBLE PRECISION array, dimension (N)   
            The scale factors for A; not accessed if EQUED = 'N'.  S is   
            an input argument if FACT = 'F'; otherwise, S is an output   
            argument.  If FACT = 'F' and EQUED = 'Y', each element of S   
            must be positive.   

    B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)   
            On entry, the N-by-NRHS right hand side matrix B.   
            On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',   
            B is overwritten by diag(S) * B.   

    LDB     (input) INTEGER   
            The leading dimension of the array B.  LDB >= max(1,N).   

    X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)   
            If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to   
            the original system of equations.  Note that if EQUED = 'Y',   
            A and B are modified on exit, and the solution to the   
            equilibrated system is inv(diag(S))*X.   

    LDX     (input) INTEGER   
            The leading dimension of the array X.  LDX >= max(1,N).   

    RCOND   (output) DOUBLE PRECISION   
            The estimate of the reciprocal condition number of the matrix   
            A after equilibration (if done).  If RCOND is less than the   
            machine precision (in particular, if RCOND = 0), the matrix   
            is singular to working precision.  This condition is   
            indicated by a return code of INFO > 0.   

    FERR    (output) DOUBLE PRECISION array, dimension (NRHS)   
            The estimated forward error bound for each solution vector   
            X(j) (the j-th column of the solution matrix X).   
            If XTRUE is the true solution corresponding to X(j), FERR(j)   
            is an estimated upper bound for the magnitude of the largest   
            element in (X(j) - XTRUE) divided by the magnitude of the   
            largest element in X(j).  The estimate is as reliable as   
            the estimate for RCOND, and is almost always a slight   
            overestimate of the true error.   

    BERR    (output) DOUBLE PRECISION array, dimension (NRHS)   
            The componentwise relative backward error of each solution   
            vector X(j) (i.e., the smallest relative change in   
            any element of A or B that makes X(j) an exact solution).   

    WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)   

    IWORK   (workspace) INTEGER array, dimension (N)   

    INFO    (output) INTEGER   
            = 0:  successful exit   
            < 0:  if INFO = -i, the i-th argument had an illegal value   
            > 0:  if INFO = i, and i is   
                  <= N:  the leading minor of order i of A is   
                         not positive definite, so the factorization   
                         could not be completed, and the solution has not   
                         been computed. RCOND = 0 is returned.   
                  = N+1: U is nonsingular, but RCOND is less than machine   
                         precision, meaning that the matrix is singular   
                         to working precision.  Nevertheless, the   
                         solution and error bounds are computed because   
                         there are a number of situations where the   
                         computed solution can be more accurate than the   
                         value of RCOND would suggest.   

    Further Details   
    ===============   

    The band storage scheme is illustrated by the following example, when   
    N = 6, KD = 2, and UPLO = 'U':   

    Two-dimensional storage of the symmetric matrix A:   

       a11  a12  a13   
            a22  a23  a24   
                 a33  a34  a35   
                      a44  a45  a46   
                           a55  a56   
       (aij=conjg(aji))         a66   

    Band storage of the upper triangle of A:   

        *    *   a13  a24  a35  a46   
        *   a12  a23  a34  a45  a56   
       a11  a22  a33  a44  a55  a66   

    Similarly, if UPLO = 'L' the format of A is as follows:   

       a11  a22  a33  a44  a55  a66   
       a21  a32  a43  a54  a65   *   
       a31  a42  a53  a64   *    *   

    Array elements marked * are not used by the routine.   

    =====================================================================   


       Parameter adjustments */
    /* Table of constant values */
    static integer c__1 = 1;
    
    /* System generated locals */
    integer ab_dim1, ab_offset, afb_dim1, afb_offset, b_dim1, b_offset, 
	    x_dim1, x_offset, i__1, i__2;
    doublereal d__1, d__2;
    /* Local variables */
    static doublereal amax, smin, smax;
    static integer i__, j;
    extern logical lsame_(char *, char *);
    static doublereal scond, anorm;
    extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, 
	    doublereal *, integer *);
    static logical equil, rcequ, upper;
    static integer j1, j2;
    extern doublereal dlamch_(char *), dlansb_(char *, char *, 
	    integer *, integer *, doublereal *, integer *, doublereal *);
    extern /* Subroutine */ int dpbcon_(char *, integer *, integer *, 
	    doublereal *, integer *, doublereal *, doublereal *, doublereal *,
	     integer *, integer *), dlaqsb_(char *, integer *, 
	    integer *, doublereal *, integer *, doublereal *, doublereal *, 
	    doublereal *, char *);
    static logical nofact;
    extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, 
	    doublereal *, integer *, doublereal *, integer *), 
	    xerbla_(char *, integer *), dpbequ_(char *, integer *, 
	    integer *, doublereal *, integer *, doublereal *, doublereal *, 
	    doublereal *, integer *);
    static doublereal bignum;
    extern /* Subroutine */ int dpbrfs_(char *, integer *, integer *, integer 
	    *, doublereal *, integer *, doublereal *, integer *, doublereal *,
	     integer *, doublereal *, integer *, doublereal *, doublereal *, 
	    doublereal *, integer *, integer *), dpbtrf_(char *, 
	    integer *, integer *, doublereal *, integer *, integer *);
    static integer infequ;
    extern /* Subroutine */ int dpbtrs_(char *, integer *, integer *, integer 
	    *, doublereal *, integer *, doublereal *, integer *, integer *);
    static doublereal smlnum;
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
#define x_ref(a_1,a_2) x[(a_2)*x_dim1 + a_1]
#define ab_ref(a_1,a_2) ab[(a_2)*ab_dim1 + a_1]
#define afb_ref(a_1,a_2) afb[(a_2)*afb_dim1 + a_1]


    ab_dim1 = *ldab;
    ab_offset = 1 + ab_dim1 * 1;
    ab -= ab_offset;
    afb_dim1 = *ldafb;
    afb_offset = 1 + afb_dim1 * 1;
    afb -= afb_offset;
    --s;
    b_dim1 = *ldb;
    b_offset = 1 + b_dim1 * 1;
    b -= b_offset;
    x_dim1 = *ldx;
    x_offset = 1 + x_dim1 * 1;
    x -= x_offset;
    --ferr;
    --berr;
    --work;
    --iwork;

    /* Function Body */
    *info = 0;
    nofact = lsame_(fact, "N");
    equil = lsame_(fact, "E");
    upper = lsame_(uplo, "U");
    if (nofact || equil) {
	*(unsigned char *)equed = 'N';
	rcequ = FALSE_;
    } else {
	rcequ = lsame_(equed, "Y");
	smlnum = dlamch_("Safe minimum");
	bignum = 1. / smlnum;
    }

/*     Test the input parameters. */

    if (! nofact && ! equil && ! lsame_(fact, "F")) {
	*info = -1;
    } else if (! upper && ! lsame_(uplo, "L")) {
	*info = -2;
    } else if (*n < 0) {
	*info = -3;
    } else if (*kd < 0) {
	*info = -4;
    } else if (*nrhs < 0) {
	*info = -5;
    } else if (*ldab < *kd + 1) {
	*info = -7;
    } else if (*ldafb < *kd + 1) {
	*info = -9;
    } else if (lsame_(fact, "F") && ! (rcequ || lsame_(
	    equed, "N"))) {
	*info = -10;
    } else {
	if (rcequ) {
	    smin = bignum;
	    smax = 0.;
	    i__1 = *n;
	    for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
		d__1 = smin, d__2 = s[j];
		smin = min(d__1,d__2);
/* Computing MAX */
		d__1 = smax, d__2 = s[j];
		smax = max(d__1,d__2);
/* L10: */
	    }
	    if (smin <= 0.) {
		*info = -11;
	    } else if (*n > 0) {
		scond = max(smin,smlnum) / min(smax,bignum);
	    } else {
		scond = 1.;
	    }
	}
	if (*info == 0) {
	    if (*ldb < max(1,*n)) {
		*info = -13;
	    } else if (*ldx < max(1,*n)) {
		*info = -15;
	    }
	}
    }

    if (*info != 0) {
	i__1 = -(*info);
	xerbla_("DPBSVX", &i__1);
	return 0;
    }

    if (equil) {

/*        Compute row and column scalings to equilibrate the matrix A. */

	dpbequ_(uplo, n, kd, &ab[ab_offset], ldab, &s[1], &scond, &amax, &
		infequ);
	if (infequ == 0) {

/*           Equilibrate the matrix. */

	    dlaqsb_(uplo, n, kd, &ab[ab_offset], ldab, &s[1], &scond, &amax, 
		    equed);
	    rcequ = lsame_(equed, "Y");
	}
    }

/*     Scale the right-hand side. */

    if (rcequ) {
	i__1 = *nrhs;
	for (j = 1; j <= i__1; ++j) {
	    i__2 = *n;
	    for (i__ = 1; i__ <= i__2; ++i__) {
		b_ref(i__, j) = s[i__] * b_ref(i__, j);
/* L20: */
	    }
/* L30: */
	}
    }

    if (nofact || equil) {

/*        Compute the Cholesky factorization A = U'*U or A = L*L'. */

	if (upper) {
	    i__1 = *n;
	    for (j = 1; j <= i__1; ++j) {
/* Computing MAX */
		i__2 = j - *kd;
		j1 = max(i__2,1);
		i__2 = j - j1 + 1;
		dcopy_(&i__2, &ab_ref(*kd + 1 - j + j1, j), &c__1, &afb_ref(*
			kd + 1 - j + j1, j), &c__1);
/* L40: */
	    }
	} else {
	    i__1 = *n;
	    for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
		i__2 = j + *kd;
		j2 = min(i__2,*n);
		i__2 = j2 - j + 1;
		dcopy_(&i__2, &ab_ref(1, j), &c__1, &afb_ref(1, j), &c__1);
/* L50: */
	    }
	}

	dpbtrf_(uplo, n, kd, &afb[afb_offset], ldafb, info);

/*        Return if INFO is non-zero. */

	if (*info != 0) {
	    if (*info > 0) {
		*rcond = 0.;
	    }
	    return 0;
	}
    }

/*     Compute the norm of the matrix A. */

    anorm = dlansb_("1", uplo, n, kd, &ab[ab_offset], ldab, &work[1]);

/*     Compute the reciprocal of the condition number of A. */

    dpbcon_(uplo, n, kd, &afb[afb_offset], ldafb, &anorm, rcond, &work[1], &
	    iwork[1], info);

/*     Set INFO = N+1 if the matrix is singular to working precision. */

    if (*rcond < dlamch_("Epsilon")) {
	*info = *n + 1;
    }

/*     Compute the solution matrix X. */

    dlacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx);
    dpbtrs_(uplo, n, kd, nrhs, &afb[afb_offset], ldafb, &x[x_offset], ldx, 
	    info);

/*     Use iterative refinement to improve the computed solution and   
       compute error bounds and backward error estimates for it. */

    dpbrfs_(uplo, n, kd, nrhs, &ab[ab_offset], ldab, &afb[afb_offset], ldafb, 
	    &b[b_offset], ldb, &x[x_offset], ldx, &ferr[1], &berr[1], &work[1]
	    , &iwork[1], info);

/*     Transform the solution matrix X to a solution of the original   
       system. */

    if (rcequ) {
	i__1 = *nrhs;
	for (j = 1; j <= i__1; ++j) {
	    i__2 = *n;
	    for (i__ = 1; i__ <= i__2; ++i__) {
		x_ref(i__, j) = s[i__] * x_ref(i__, j);
/* L60: */
	    }
/* L70: */
	}
	i__1 = *nrhs;
	for (j = 1; j <= i__1; ++j) {
	    ferr[j] /= scond;
/* L80: */
	}
    }

    return 0;

/*     End of DPBSVX */

} /* dpbsvx_ */

#undef afb_ref
#undef ab_ref
#undef x_ref
#undef b_ref


